Remark 5.10  By the convexity property of \({\bf M}\)  established in Lemma REF  (and using that \({\mathcal {S}}({\mathrm {P}}, {\mathrm {L}})\)  is linearly convex) for given \(A,B\) , there exists at most one, up to an addition with an affine-linear function, \(u\in {\mathcal {S}}({\mathrm {P}}, {{\mathrm {L}}})\)  which satisfies \(\kappa (u, A, B)=a\) . One can also show that for any \(u\in {\mathcal {S}}({\mathrm {P}}, {{\mathrm {L}}})\) , the corresponding \({\bf X}\)  is a smooth, matrix-valued function on \({\mathrm {P}}\)  which satisfies the positivity and boundary conditions of , and thus (REF ) gives rise to an almost Kähler structure with constant scalar curvature in the sense of [1]}. The necessary condition for the existence of such \(u\)  is the one of uniform relative K-stability of \(({\mathrm {P}}, {{\mathrm {L}}})\)  (see [2]}, [3]}, [4]}) which implies that there exist constants \(\lambda , \delta >0\)  such that
\({\bf F}(u) \ge \lambda \int _{{\mathrm {P}}} |\mathring{u}| {\mathrmsl d}x -\delta , \qquad \forall u \in {\mathcal {S}}({\mathrm {P}}, {{\mathrm {L}}}),\) 
